December 6, 2021

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A Grad Student Solved the Epic Conway Knot Problem—in a Week

In the summer of 2018, at a conference on low-dimensional topology and geometry, Lisa Piccirillo heard about a nice little math problem. It seemed like a good testing ground for some techniques she had been developing as a graduate student at the University of Texas at Austin.

“I didn’t allow myself to work on it during the day,” she said, “because I didn’t consider it to be real math. I thought it was, like, my homework.”

The question asked whether the Conway knot—a snarl discovered more than half a century ago by the legendary mathematician John Horton Conway—is a slice of a higher-dimensional knot. “Sliceness” is one of the first natural questions knot theorists ask about knots in higher-dimensional spaces, and mathematicians had been able to answer it for all of the thousands of knots with 12 or fewer crossings—except one. The Conway knot, which has 11 crossings, had thumbed its nose at mathematicians for decades.

Before the week was out, Piccirillo had an answer: The Conway knot is not “slice.” A few days later, she met with Cameron Gordon, a professor at UT Austin, and casually mentioned her solution.

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“I said, ‘What?? That’s going to the Annals right now!’” Gordon said, referring to Annals of Mathematics, one of the discipline’s top journals.

“He started yelling, ‘Why aren’t you more excited?’” said Piccirillo, now a postdoctoral fellow at Brandeis University. “He sort of freaked out.”

“I don’t think she’d recognized what an old and famous problem this was,” Gordon said.

Piccirillo’s proof appeared in Annals of Mathematics in February. The paper, combined with her other work, has secured her a tenure-track job offer from the Massachusetts Institute of Technology that will begin on July 1, only 14 months after she finished her doctorate.

The question of the Conway knot’s sliceness was famous not just because of how long it had gone unsolved. Slice knots give mathematicians a way to probe the strange nature of four-dimensional space, in which two-dimensional spheres can be knotted, sometimes in such crumpled ways that they can’t be smoothed out. Sliceness is “connected to some of the deepest questions in four-dimensional topology right now,” said Charles Livingston, an emeritus professor at Indiana University.

“This question, whether the Conway knot is slice, had been kind of a touchstone for a lot of the modern developments around the general area of knot theory,” said Joshua Greene of Boston College, who supervised Piccirillo’s senior thesis when she was an undergraduate there. “It was really gratifying to see somebody I’d known for so long suddenly pull the sword from the stone.”

Magic Spheres

While most of us think of a knot as existing in a piece of string with two ends, mathematicians think of the two ends as joined, so the knot can’t unravel. Over the past century, these knotted loops have helped illuminate subjects from quantum physics to the structure of DNA, as well as the topology of three-dimensional space.

John Conway in 1990 explaining how in high school he showed why two knots can’t cancel each other out. 

But our world is four-dimensional if we include time as a dimension, so it is natural to ask if there is a corresponding theory of knots in 4D space. This isn’t just a matter of taking all the knots we have in 3D space and plunking them down in 4D space: With four dimensions to move around in, any knotted loop can be unraveled if strands are moved over each other in the fourth dimension.

To make a knotted object in four-dimensional space, you need a two-dimensional sphere, not a one-dimensional loop. Just as three dimensions provide enough room to build knotted loops but not enough room for them to unravel, four dimensions provide such an environment for knotted spheres, which mathematicians first constructed in the 1920s.